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  4. tan-1[cos(2 tan-1 (3/4)) +sin(2 cot-1 (1/2))] is

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Q. $tan^{-1}\left[cos\left(2\,tan^{-1} \frac{3}{4}\right) +sin\left(2\,cot^{-1} \frac{1}{2}\right)\right] $ is

Inverse Trigonometric Functions

Solution:

$cos\left(2\,tan^{-1} \frac{3}{4}\right) = \frac{1-\frac{9}{16}}{1+\frac{9}{16}} = \frac{7}{25}$
$sin\left(2\,cot^{-1} \frac{1}{2}\right) = sin\left(2\,tan^{-1}\,2\right)$
$= \frac{2\times2}{1+4} = \frac{4}{5}$
$tan^{-1}\left(\frac{7}{25}+\frac{4}{5}\right)$
$= tan^{-1} \frac{27}{25} > \frac{\pi}{4}$